Have you ever applied scale factor to a real-world dilemma? Take a look at this one.

A driveway has a length of 24 feet. If the scale is 2 inches : 4 feet, what is the scale factor? In a diagram, how many inches would be drawn to represent the driveway?

Pay attention and you will know how to figure this out by the end of the Concept.

Guidance

A
ratio
is a comparison between two quantities. We can write a ratio in fraction form, by using a colon or by using the word “to”.

Sometimes in life, we have a real-life object that we want to represent in a smaller form. Think about buildings. We can’t build an actual building to show the dimensions in a smaller way, so we build a model of the building. When we do this, we take the actual dimensions and shrink them down to build a model.

The scale that we use can help us with scale dimensions or actual dimensions. This scale is key in problem solving.

Let’s say that the scale is 1 : 2.

We can use this information to determine the
scale factor
. The
scale factor
is the relationship between the scale dimension and the measurement comparison between the scale measurement of the model and the actual length.

In this case, it is
.

Take a look at this situation where we can use scale factor.

What is the scale factor if 3 inches is equal to 12 feet?

We can write a ratio to show the scale factor.

The scale factor is 1 : 4. It is expressed in simplest form.

Now let’s look at applying this information further.

If the scale dimension is 4, then we can figure out the actual dimension. Here is a proportion to show these two ratios.

See the units aren’t necessary for figuring out the missing part of the proportion. We can simply use what we have learned to find the actual dimension.

1 times 4 = 4

2 times 4 = 8

This is the answer.

Now we can look at applying scale factor to our work when we do know the units. To use scale factor to find actual dimensions or scale dimensions, we will need to know a few things.

Necessary Information:

Scale Factor

One other dimension either the actual or the scale dimension must be given

So, if we have three parts of the proportion, we can solve for the last missing part.

Take a look at this one.

The plans for a flower garden show that it is 6 inches wide on the plan. If the scale for the flower garden is 1 : 12, what is the actual width of the flower garden?

To work on this problem, we first need to write two ratios that form a proportion. We have the scale factor and we have the scale measurement. We are missing the actual measurement. Let’s figure out the actual measurement of the garden.

Now we have two ratios that form a proportion. Let’s write them both in fraction form so that we can work easily in solving for the missing measurement.

Now we can cross multiply or solve it by using equal ratios.

The measurement of the garden is 36 inches, which is the same as three feet.

Use the scale factor of
":
' to find the actual dimensions in each example.

Example A

"

Solution:
inch

Example B

"

Solution:
inch

Example C

"

Solution:
inch

Now let's go back to the dilemma from the beginning of the Concept.

Notice that there are two parts to this problem. First, we have to identify the scale factor.

The scale factor is 1 : 2.

Next, we need to figure out how many inches will be drawn to represent the driveway. To do this, we write a proportion.

We can cross multiply and divide or use equal ratios to solve this. Let’s use equal ratios. We work with the denominators.

The driveway will be represented by 12 inches or 1 foot.

Guided Practice

Here is one for you to try on your own.

Find the missing actual dimension if the scale factor is 2" : 3' and the scale measurement is 6".

Solution

First, we can set up a proportion.

Now we can use fraction form to make it easier to solve this proportion.